This is a research announcement on what is best termed `nonlocal' methods in
mathematics. (This is not to be confused with global analysis.) The nonlocal
formulation of physics in \cite{principia} points to a fresh viewpoint in
mathematics: Nonlocal viewpoint. It involves analyzing objects of geometry and
analysis using nonlocal methods, as opposed to the classical local methods,
e.g., Newton's calculus. It also involves analyzing new nonlocal geometries and
nonlocal analytical objects, i.e. nonlocal fields. In geometry, we introduce
and study (nonlocal) forms, differentials, integrals, connections, curvatures,
holonomy, G-structures, etc. In analysis, we analyze local fields using
nonlocal methods (semilocal analysis); nonlocal fields nonlocally (of course);
and the connection between nonlocal linear analysis and local nonlinear
analysis. Analysis and geometry are next synthesized to yield nonlocal (hence
noncommutative) homology, cohomology, de Rham theory, Hodge theory, Chern-Weil
theory, K-theory (called N-theory) and index theory. Applications include
theorems such as nonlocal-noncommutative Riemann-Roch, Gauss-Bonnet, Hirzebruch
signature, etc. The nonlocal viewpoint is also investigated in algebraic
geometry, analytic geometry, and, to some extent, in arithmetic geometry,
resulting in powerful bridges between the classical and nonlocal-noncommutative
aspects.